3.
Joule Dissipation: Order of Magnitude Estimates

The discussion in the previous section indicates that there are two
limits which we can apply to the Venus ionosphere. The first is
when electron-ion collisions dominate, while the second is when
electron-neutral collisions dominate. As noted earlier, the former is
more appropriate for high altitudes, while the latter is more
appropriate for the bottomside ionosphere.

When electron-ion collisions dominate, we can rewrite the
heating and cooling rates, expressed in W/m, as:

(9)

(10)

(11)

In (9), L is the scale length for heat conduction, as discussed
below, given in kilometers. In (10), E is the wave electric field in
volts per meter, B
is the ambient magnetic field strength in
nanoteslas, and we have assumed that the wave is polarized
perpendicularly to the ambient magnetic field. The term in
parentheses in (10) ~ 1 when
v
<<
, as is usually the case when
electron-ion collisions dominate. In (11),
m and
m
are the ion
and proton masses respectively, and we have assumed that
T>>
T.
In all these equations
T
is in eV, densities are in
cm, and
is
again the Coulomb logarithm.

In specifying a scaling law for the heat conduction, from
(2)
·q -
(KT) = -
(KT
+KT).
When electron-ion
collisions dominate, KT and is
independent of n.
Hence the first term in parentheses will always
be positive, and we therefore require
T
< 0 for
· q > 0.
This condition
is not satisfied for a temperature dependence such as
T exp(-x/L),
but it is for T
exp(-x /L)
when x ~ 0. In deriving a scaling law we therefore assume that the
temperature profile is such that
·q
> 0, as required for the electron
energy budget, and simply take
-
(KT
)= K
T/
L,
where L is a heat conduction
scale length.

When electron-neutral collisions dominate,

(9')

(10')

(11')

The symbols have the same meaning as in (9) to (11), and have the
same units. In (9'), is
the electron-neutral collisional cross
section, and
is the cross section for oxygen = 2
10
cm.
When electron-neutral collisions dominate, the thermal
conductivity tends to increase with altitude, and a negative
temperature gradient will generally ensure that
·q > 0. In (10') the
term in parentheses ~1 when
v
>>
. In (11') m is the neutral
mass, and we have assumed
T >>
T.

Turning first to the dayside ionopause,
Figure 2 shows the
various heating and cooling rates as a function of electron density
(Figure 2a)
and electron temperature (Figure 2b).
We assume that
the heat conduction scale (L) is 1000 km, the ambient magnetic
field strength (B)
is 50 nT, and the wave electric field amplitude
(E) is 10 mV/m. Also, as noted above, we assume that electron-ion
collisions dominate, i.e., we are using
(9) to (11),
and the ions are
O
[e.g., Brace and Kliore, 1991]. We choose a scale length of
1000 km since the ambient magnetic field is draped over the
ionopause, and the important scales are horizontal. These scales
will be much longer than vertical scales, and 1000 km is a
reasonable order of magnitude estimate. Above the dayside
ionopause, wave spectral amplitudes can be as large as
10 V/m/
Hz
[Strangeway, 1991a]. We convert the spectral amplitude
to a wave electric field amplitude by assuming that the spectral
bandwidth is of order the wave frequency. Thus, at 100 Hz a
spectral amplitude of 10
V/m/Hz corresponds roughly to
an electric field amplitude of 10 mV/m.

Figure 2.
Electron heating and cooling rate estimates for the
dayside ionopause. Since electron-ion collisions dominate, the
rates are given by (9) to
(11).
The rates are shown (a) as a function
of density for T =
1 eV and (b) as a function of temperature for
n
= 1000 cm.
It is also assumed that L = 1000 km,
B = 50 nT, E
= 10 mV/m, and the ions are
O.
As an indication of the relative
importance of the heating and cooling rates, we include the energy
density (nT).
The dot on this line marks
= 1, where we expect
kinetic effects to be important.

In Figure 2a
we assume that T
= 1 eV, which is large for
the ionosphere but is appropriate for the transitional region where
the waves are observed. It is clear that for fixed temperature the
Joule dissipation is always greater than the elastic collisional
cooling. This is because both have the same dependence on the
electron-ion collision frequency, and hence on electron density.
More importantly, the electron heat conduction exceeds the Joule
dissipation by a sufficiently large factor that we can increase the
conduction scale length to 3000 km and still match the Joule
dissipation, even for the highest densities shown. For densities of
1000 cm
the conduction scale could be as large as 30,000 km,
about 5 Venus radii.

We also show the energy density of the electrons,
nT, to
indicate how important the different heating and cooling terms are,
since nT
divided by the cooling or heating rate gives the
approximate cooling/heating time constant. It is clear that except at
the higher densities the Joule dissipation is relatively weak, with a
time constant of more than 100 s. On the
nT curve we mark
where = 1, indicated by the dot.
Thus at the higher densities we
might expect kinetic effects to be more important, possibly
enhancing the dissipation rate. However, heat conduction still
plays a significant role in the electron heat budget.

Figure 2b
shows how the rates depend on electron
temperature. For
Figure 2b we assume that
n = 1000
cm. For
very low temperatures the Joule dissipation can exceed the heat
conduction. However, (9)
shows that the heat conduction has a
strong dependence on temperature, while (10)
shows that the Joule
dissipation decreases with increasing temperature. Thus even
though the Joule dissipation may initially exceed the heat
conduction, a small increase in temperature is sufficient to match
Joule dissipation by heat conduction.
Figures 2a and 2b
demonstrate that Joule dissipation through electron-ion collisions is
not an important source of heating for the dayside ionopause.

At higher altitudes within the nightside ionosphere
electron-ion collisions will usually dominate, except for the lower
plasma densities. The heating and cooling rates for this case are
shown in
Figure 3a. For
Figure 3a we assume that L = 10 km,
B =
30 nT, T =
0.1 eV, and E = 1 mV/m. These parameters have been
chosen to correspond to an ionospheric hole [Brace and Kliore,
1991]. We have assumed that the ambient ions are
O, which is
usually the case at altitudes
150 km [Grebowsky et al., 1993],
while we also assume that the neutrals are atomic oxygen with a
density of 4
10.
Since the magnetic field within a hole is
generally radial, it is appropriate to consider vertical scales, and we
choose a scale of 10 km. Also, since the waves tend to be
somewhat weaker in amplitude, we assume a wave electric field of
1 mV/m. This wave amplitude corresponds to a Poynting flux of ~
3 10
W/m
for a refractive index ~ 1000, Russell et al. [1989b]
reported a median Poynting flux of ~ 10 W/m, assuming a 30-Hz bandwidth.

For the particular choice of wave and plasma parameters in
Figure 3a
, we again find that electron heat conduction can easily
match the Joule dissipation within an ionospheric hole. Even
though the heat conduction is of the same order as the Joule
dissipation for densities of 10
cm, we do not expect the wave
amplitudes to be as high as 1 mV/m for these densities. At these
high densities the scaling laws are given by
(9) to (11), while at
lower electron densities (9') to
(11') apply. Whistler mode waves
tend not to be observed within the high-density regions at higher
altitudes, since they are Landau damped and gyrodamped
[Strangeway, 1992; 1995a]. The conduction scale length can be as
large as 50 km for densities of 10
cm, and the conduction
cooling will still exceed the Joule dissipation. Thus similarly to the
dayside ionopause, we do not expect collisional Joule dissipation
to be important at moderate altitudes (
150 km) within the
nightside ionosphere.

In Figure 3b
we have chosen wave and plasma parameters
corresponding to the bottomside ionosphere. We assume that L = 2
km, B
= 5 nT, n
= 1000 cm, E =
10 mV/m, the neutrals are CO
and
N
= 10 cm.
The wave amplitude corresponds to
the wave intensities observed on the very low altitude passes
during the Pioneer Venus Orbiter entry phase [Strangeway et al.,
1993b]. We have chosen the very short scale length of 2 km, since
this is of order the attenuation scale observed for the 100 Hz waves
[Strangeway et al., 1993b], and is also of order the density scale
height for CO
[Kasprzak et al., 1993]. However,
Figure 3b shows
that even for this short a scale the Joule dissipation exceeds the
conduction cooling, except for the higher temperatures. Thus we
might expect that collisional Joule dissipation is an important heat
source for the bottomside ionosphere.

In addition to the elastic cooling rate, we have also included
the cooling rate through vibrational excitation of
CO in
Figure 3b.
This cooling rate often exceeds that due to electron heat
conduction, and may in fact be the means for balancing the Joule
dissipation at the lowest altitudes. However, as discussed in the
appendix, this cooling operates whether or not waves are present,
and the cooling rate is so large that it may have important
implications for the electron heat budget. In the absence of any
other heat source, the vibrational cooling must be balanced by the
heat conduction into the volume. Taking the heat conduction curve
in
Figure 3b as a guide, it is clear that the temperature gradient
scale must be very short to supply sufficient heating, and as shown
in the appendix, large topside temperatures may be required to
provide the downward heat flux necessary to balance the
vibrational cooling at the bottomside.

Last, in
Figure 3b,
> 1. However, since the collision
frequency is >>
and ,
it is by no means clear that resonant
wave-particle effects are important. Electron motion is almost
certainly dominated by collisions, and it is unlikely that electrons
can remain in resonance with waves.

In concluding from
Figure 3b that wave Joule dissipation is
important for the bottomside we used a fixed wave amplitude and
fixed neutral and plasma density. However, all these parameters
are changing on very short vertical scales within the bottomside
ionosphere. The Joule dissipation that causes electron heating also
reduces the wave energy. If the heating rate is high, we would
expect that very little wave energy would propagate into the
ionosphere. It is therefore necessary to take into account the
variation of the waves and the ambient neutrals and plasma if we
are to assess realistically the importance of collisional Joule
dissipation as a heat source for the ionosphere.